How to determine if the set of vectors are linearly dependent or independent

Question

Determine if the following sets of vectors are linearly dependent or linearly independent

$$V1=\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0\end{bmatrix}$$ $$V2=\begin{bmatrix}0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}$$ $$V3=\begin{bmatrix}0 & 0 & 0 \\0 & 1 & 0\end{bmatrix}$$

I'm not sure how to solve this as previous examples I have always just worked out the determinant but I can't for this. How do I go about solving this? Thanks

Answer 1

Use the definition. A set of vectors is linearly dependent if a nonzero combination of them results in the zero vector.

http://en.wikipedia.org/wiki/Linear_independence#Definition

Answer 2

Just use the definition. Consider the equation $$\alpha V_1 + \beta V_2 + \gamma V_3 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \; ,$$ and show, that this equation only holds iff $\alpha = \beta = \gamma = 0$.

Answer 3

In this case, it's easy to see that $V_{1},V_{2}$ and $V_{3}$ are independent because you can simply observe that no one of them is a linear combination of the others.